High-precision ghz clock generation using spin states in diamond

ABSTRACT

Techniques for obtaining a frequency standard using the crystal field splitting frequency of nitrogen vacancy center in diamond are disclosed. In certain exemplary embodiments, a microwave field is applied to the diamond and optically exciting the diamond under green light. The photoluminescent response of the diamond is measured by a photodetector. The intensity of the photoluminescent response can be used to determine the phase shift between the microwave and the crystal field splitting frequency. The microwave field frequency can be adjusted until the phase shift is below a predetermined threshold, and the microwave frequency can then be output for use as a standard.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application Ser.No. 61/535,808, filed Sep. 16, 2011, and U.S. Provisional ApplicationSer. No. 61/693,416, filed Aug. 27, 2012 the disclosures of which areeach hereby incorporated by reference in their entirety.

STATEMENT OF FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under Grant No.FA9550-11-0014, awarded by the Air Force Office of Scientific Research.The government has certain rights in the invention.

BACKGROUND

The disclosed subject matter relates to techniques for providing afrequency standard using spin states of a defect in a diamond structure.

Atomic clocks can serve as the basis for accurate systems in use formeasuring time and frequency. They can be utilized in a number ofapplications, including communication, computation, mobile devices,sensors, autonomous vehicles, undersea oil/gas exploration, spacenavigation, aviation, cruise missiles, and navigation systems (e.g.global positioning systems (GPS)).

Certain frequency standards can derive their stability from hyperfinelevel splitting of energy states in atoms such as Cs, Rb, or H. When anoscillating magnetic field is resonant with the energy difference ofthese internal states, a change in population between levels changes theradiofrequency or optical absorption. Certain techniques can modulatethe driving frequency and monitor the absorption as a correction for atunable active reference oscillator, e.g. a quartz crystal, thusstabilizing it to the atomic line.

Single trapped ions and ensembles of atoms trapped in optical latticescan provide a frequency standard with accuracy exceeding theinternational cesium standard, and can enable the observation of generalrelativity corrections within a few meters. Such techniques, however,can require infrastructure that encompasses several tens of cubic metersof space.

Portable standards based on rubidium vapor cells can provide stabilityfor time scales ranging from 1 s to 10⁴ s and can be used in connectionwith satellites, laboratory equipment, and cellular communications.Mobile devices, which may not contain their own precision standards, canshare GPS time signals for maintaining communication standards, but whenthe external lock signal is obstructed, a precise local frequencystandard with minimal drift can be necessary to maintainsynchronization.

Accordingly, there is a need for improved techniques for providing afrequency standard for time-keeping.

Nitrogen Vacancy Centers (NVC) are point defects in the diamond crystalstructure. The substitution of a nitrogen atom for a carbon atom in thediamond structure can create a lattice vacancy that can be occupied bythree unpaired electrons. In a neutral nitrogen vacancy defect, N—V⁰,two of the unpaired electrons can form a quasi covalent bond, while thethird electron remains unpaired. However, the three electrons canexhibit axial symmetry, and the three continuously exchange roles. Anegative nitrogen vacancy, N—V′, can have an additional electronassociated with the vacancy which forms an S=1 structure that has along-lived spin triplet in its ground state that can be probed usingoptical and microwave excitation. This can allow for spin manipulationof the NVC.

The NVC can have trigonal C₃₀ symmetry and 3A² ground state with totalelectronic spin S=1. Spin-spin interaction can lead to a zero-fieldsplitting between the ms=0 and ms=±1 manifolds, where the quantizationaxis is along the NV-axis. NVCs can have a crystal field splittingfrequency (D_(gs)) of, for example, 2.870 GHz in the absence of anexternal magnetic field and in the absence of other environmentalfactors, including strain. This zero-field splitting frequency can bechanged by the application of magnetic fields in the direction of theNVC through the Zeeman effect.

SUMMARY

In one aspect of the disclosed subject matter, a method of obtaining afrequency standard from the crystal field splitting frequency of anitrogen vacancy center in a diamond structure is provided. In oneembodiment, the method can include applying an initial, adjustablemicrowave field to a diamond structure. Optical excitation of thediamond structure can produce a detectable photoluminescent response,with that response being modulated in intensity by the frequency of themicrowave field being applied. From the modulation of thephotoluminescent response, the phase shift between the microwave fieldand the crystal field splitting frequency of the nitrogen vacancy centerin the diamond structure can be determined. The frequency of themicrowave field can be adjusted until the detected phase shift is belowa predetermined threshold. That microwave frequency then can be used asthe frequency standard.

In certain embodiments, the microwave field frequency can be testedcontinuously, even after the original determination of the frequencystandard, to prevent drift of the frequency standard.

In one embodiment, an ensemble of nitrogen vacancy centers can beoptically excited and probed with the microwave field to find thefrequency standard. The use of multiple nitrogen vacancy centers canprovide additional accuracy in the measurement of the frequencystandard.

In another embodiment, a sequence of light and microwave pulses can beused. For example, the diamond structure can be first subjected tooptical pumping to drive the spin state of a plurality of nitrogenvacancy centers to the |0> state. After optical pumping ceases, a firstmicrowave pulse of flip angle π/4 can be applied to the diamondstructure. After a time T, a second microwave pulse of flip angle π canbe applied in the opposite phase of the first pulse. After a second timeT, a third microwave pulse identical to the first microwave pulse can beapplied. Following the third microwave pulse, the transient fluorescentresponse of the diamond structure can be measured by applying an opticalpulse. The measurement of the transient fluorescent response can be usedto calculate the phase shift of the initial microwave pulse frequencyfrom the crystal field splitting frequency.

In an embodiment, two diamond structures can be tested at the same timein order to stabilize the measurements against temperature dependence.In another embodiment, a single diamond structure can be placed in aclamp to engineer strain upon the diamond structure, in order tostabilize the measurements against temperature dependence.

In another aspect of the disclosed subject matter, a system forobtaining a frequency standard based on the crystal field splittingfrequency of nitrogen vacancy centers in a diamond structure isprovided. One exemplary system can include a dielectric cavity adaptedto at least partially encompass the diamond structure, a light sourceconfigured to optically excite the nitrogen vacancy center and therebyallow the nitrogen vacancy center to emit a photoluminescent response, aphotodetector configured to detect photoluminescence, a striplineconfigured to emit a microwave field upon the diamond structure, aprocessor to calculate the phase shift between the microwave emissionfrequency and the crystal field splitting frequency of the nitrogenvacancy center in the diamond structure, and a controller to adjust thefrequency of the microwave emissions.

In an embodiment, the photodetector can be a silicon photodiode. Thedielectric cavity can include a pair of Bragg reflectors placed onopposite sides of the diamond structure and outside the plane of thestripline for microwave emissions. The Bragg reflectors can create a 532nm resonant cavity encompassing the diamond structure. The light sourcecan be a surface emitting laser adapted for generation of a doubled 1064nm beam.

In an embodiment, the light source and the microwave stripline can beconfigured to apply pulsed emissions upon the diamond structure. Thephotodetector can be configured to measure the photoluminescent responseafter an optical pulse.

In an embodiment, two diamond structures can be placed in two systems inthe dielectric cavity, the light source, the photodetector, and thestripline. The processor can be configured to detect the phase shift forboth diamond structures and thereby stabilize the measurements fortemperature dependence. In a further embodiment, the two diamondstructures can be clamped to two substrates with different thermalexpansion coefficients. The diamond structure can be coupled to a singleclamp to engineer strain upon the diamond structure.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a system for obtaining a frequencystandard from spin states of defects in a diamond structure according toan embodiment of the disclosed subject matter.

FIG. 2 is a flow diagram of a method for obtaining a frequency standardfrom spin states of defects in a diamond structure according to anembodiment of the disclosed subject matter.

FIG. 3 is a flow diagram of a method for obtaining a frequency standardfrom spin states of defects in a diamond structure according to anotherembodiment of the disclosed subject matter.

FIG. 4 is a depiction of a system for obtaining a frequency standardfrom spin states of defects in a diamond structure according to anembodiment of the disclosed subject matter.

FIG. 5( a) is a diagram of certain spin sublevels for a nitrogen vacancycenter in diamond.

FIG. 5( b) is a graph of the steady-state fluorescence emission of anitrogen vacancy center under continuous optical and microwaveirradiation.

FIG. 6 is a diagram of a method for obtaining a frequency standard fromspin states of defects in a diamond structure by exposing the diamondstructure to pulses of optical and microwave emissions, and the expectedspin states of the defect during the time the method is implemented.

FIG. 7 a is a chart illustrating the different temperature dependencesfor two different clamp materials.

FIG. 7 b is a diagram of the method for compensating for temperaturedependence using two clamps.

FIG. 7 c is a diagram of the single-clamp method for controllingtemperature dependence with engineered strain.

FIG. 8 a is depicts simulation of the field splitting frequency of anitrogen vacancy center as a function of temperature where the diamondstructure has been clamped with a stiff material and where the diamondstructure has been clamped with a softer material in accordance with anembodiment of the disclosed subject matter.

FIG. 8 b depicts the relationship between spatial frequency variationand position within a clamped diamond disk in accordance with andembodiment of the disclosed subject matter.

FIG. 8 c depicts a diamond disk and a clamp in accordance with anembodiment of the disclosed subject matter.

Throughout the drawings, the same reference numerals and characters,unless otherwise stated, are used to denote like features, elements,components or portions of the illustrated embodiments. Moreover, whilethe disclosed subject matter will now be described in detail withreference to the Figs., it is done so in connection with theillustrative embodiments.

DETAILED DESCRIPTION

Techniques for providing a frequency standard using spin states of adefect in a diamond structure are disclosed herein. Diamond structurescan have many desirable characteristics for providing a frequencystandard. For example, a single crystal diamond can be grown into amicron-scale, radiation hard chip, which makes it portable andwell-suited for integration in a semiconductor fabrication process.Additionally, a frequency standard derived from the spin lifetime of thenitrogen vacancy center (NVC) can resemble atomic and molecular systems.The optical detection of the NVC can also increase the signal-to-noiseratio for a solid-state standard. A sufficiently high density of NVCs ina chip can allow for comparable frequency stability in smaller sensorvolumes.

In one aspect of the disclosed subject matter, techniques for obtaininga frequency standard from the crystal field splitting frequency of anitrogen vacancy center in a diamond structure can include continuouslyapplying an initial, adjustable microwave field to a diamond structureconcurrently with optical excitation to produce a detectablephotoluminescent response. In another aspect of the disclosed subjectmatter, techniques for obtaining a frequency standard from the crystalfield splitting frequency of a nitrogen vacancy center in a diamondstructure can include a pulsed technique, applying one or more microwavepulses to a diamond structure, and then observing the transientphotoluminescent response by applying an optical pulse. In yet anotheraspect of the disclosed subject matter, techniques for obtaining afrequency standard from the crystal field splitting frequency of anitrogen vacancy center in a diamond structure can include usingtemperature stabilization techniques to improve the accuracy of thefrequency standard.

An exemplary embodiment of a system for obtaining a frequency standardusing spin states of defects in diamond will now be described withreference to FIG. 1. and FIG. 4, for purposes of illustration and notlimitation.

A diamond chip containing NVCs 110 on a substrate 119 can be exposed tooptical and microwave emissions. The diamond chip can be grown onto thesubstrate, for example, via chemical vapor deposition (CVD).Alternatively, a pre-fabricated diamond chip can be implanted on orintegrated with the substrate. The photoluminescent (PL) response fromthe optical excitation can be detected using a photodetector 140 and thephase shift between the microwave emission and the field-splittingfrequency, D_(gs), of the NVC can be determined using a processor 150.

The optical emissions can be created by a light source 120 arranged toilluminate one face of the diamond chip. The light source can beconfigured to produce, for example a laser beam 125 having a specificpower level 121 and a specific wavelength 123. In one embodiment, thelight source can be configured to produce a 1064 nm laser that isdoubled prior to interacting with the diamond chip 125. Alternatively,the light source can be configured to produce approximately a 520-550 nmlaser, and in certain embodiments can be configured to produce a 520 nmlaser.

A dielectric cavity 115 can be placed around the diamond chip 110 tolower the power requirements for the laser beam 125. The dielectriccavity can be formed for example by a pair of Bragg reflectors, whichcan create a 532 nm resonant cavity around the diamond chip 110. One ofthe Bragg reflectors can be placed between the diamond chip 110 and thelight source 120 and the second reflector can be placed on the oppositeside. The Bragg reflectors can be made of a quarter wave stack ofmaterials with different reflective indices, using for example,materials such as silicon dioxide, titanium dioxide, tantalum pentoxide,or the like. This can allow the laser beam 125 to resonate within thecavity and can cause optical excitation of the NVCs in the diamond chip110.

The microwave field can be created by a planar stripline 131 that canencircle the diamond chip 110 on a single plane as seen in FIG. 4.Alternatively, the microwave field can be created by other suitabletechniques, for example with other geometries and suitable antennas. Themicrowave cavity can also be fitted with reflectors to create a resonantcavity 111 for microwave emissions, for example at 2.87 GHz. A frequencycontroller 160 can determine the frequency of the microwave emissionsbased on input data from a microprocessor 150, and can report thefrequency back to the microprocessor 151.

The optical emissions can excite the NVCs, to produce a photoluminescent(PL) response 127. NVCs can absorb green light and emit a PL responsebetween, for example, 637-800 nm. As described in more detail below, theintensity of the PL response can correspond to the polarization of thespin states of the NVC, which can depend on the frequency of themicrowave field. Thus, the processor 150 can determine, based on asignal from the photodetctor 140, when the PL response corresponds to anapplied microwave field equal to a known field-splitting frequency ofthe NVC, and thus can lock the frequency controller 160 at such afrequency, thereby providing a frequency standard.

Exemplary embodiments of techniques for providing a frequency standardusing spin states of a defect in diamond will be described in detailbelow. For purposes of illustration, and not limitation, the levelstructure and fluorescent properties of the nitrogen vacancy, as theyrelate to exemplary and non-limiting bases for a frequency standarddisclosed herein (e.g., the zero-field splitting of the NV as afrequency standard), will first be described. However, one or ordinaryskill in the art will appreciate that certain properties of the NVcenter in diamond are well known, and the following description is notintended to limit or otherwise narrow the scope of the disclosed subjectmatter. Additionally, throughout the following description, certaincharacterizations of stability and quality factor will be described forpurposes of illustration, and not limitation. One of ordinary skill inthe art will appreciate that such characterizations are not intended tolimit or otherwise narrow the scope of the disclosed subject matter, butare provided as an exemplary method of describing the techniquesdisclosed herein.

Atomic clocks, generally, can drive their stability from a large qualityfactor (“Q-factor”), which can be given as Q=v/Δv, of the probedresonance, with narrow linewidth, Δv, being smaller than the resonantfrequency v. The NYC in diamond can have a ground state spin tripletcharacterized by long coherence times (for example, greater than 1 ms),a ground state crystal field splitting with an intrinsic resonancefrequency near 2.870 GHz, which can be independent, to lowest order, ofapplied magnetic field, and spin states that are optically polarizableand detectable on single defect length scales (for example,approximately 1 μm).

With reference to FIG. 5( a), relevant spin sublevels of the NVC (0, 1,2, 3 and S) are depicted. Optical absorption of green laser light cancause broadband photoluminescence (PL) of the NVC from, for example,637-800 nm. A spin dependent intersystem crossing between the excitedspin triplet 3 and the metastable, dark singlet level S can change theintegrated PL for the spin states |0> and |±1

. The deshelving from the singlet can occur primarily to the |0> spinstate, providing a means to polarize the NVC. Microwave fields resonantbetween levels |0> and |1> can perturb the spin populations, and thusthe PL response. This can be characterized, for example, in two ways: 1)by measuring a continuous wave response to simultaneous optical andmicrowave fields (for example, as demonstrated in FIG. 5( b), or 2) in apulsed manner, by preparing a state using only microwaves, and observingthe transient PL response.

For purposes of illustration, and not limitation, the relevant spindynamics can be described by Hamiltonians for the lowest and firstexcited triplet state and two metastable singlet states. In an exemplaryembodiment, as described below, the response of only the ground statetriplet sublevels to resonant excitation can be monitored, yielding theground state Hamiltonian:

H _(gs)=(D _(gs) +d∥σ _(z))S _(z) ² +gμ _(b) {right arrow over(S)}·{right arrow over (B)}+d ₁₉₅σ_(⊥)(S _(x) S _(y) +S _(y) S _(z))+d_(⊥)σ_(⊥)(S _(z) ² −S _(y) ²)   (1)

where d_(¶,⊥) are the ground state electric dipole moment componentsalong and perpendicular to the C₃₀ symmetry axis of the defect. D_(gs)is the ground state crystal field splitting frequency (e.g., 2.870 GHz),μ_(b) is the Bohr magnetron, and g is the Uncle factor (assumed to beisotropic). S_(k) are spin-1 operators in the k={x, y, z} directions.The local electric field vector, induced by crystal strain, is {rightarrow over (σ)}. In the limit of static magnetic and electric fieldsmuch smaller than D_(gs), the eigenfimctions are those of the S_(z)operator, as shown in FIG. 5.

A driving field at frequency co can induce electron spin resonance (ESR)transitions between |0> and |±1>. On resonance (ω≈D_(gs)), the PLresponse can decrease and can provide a feedback signal as to lock co toD_(gs). The dynamics can be viewed as a response to a time-varyingmagnetic field B₁=2b₁ cos(2πωt){circumflex over (x)}; transformingH_(gs) into an interaction frame defined by the operator V=e^(2xiωS)_(z) ^(is z) ² . Under the rotating wave approximation, the Hamiltoniancan be:

H _(gs) ¹=(D _(gs) +d∥σ_(z)−ω)S _(z) ² +g μ _(b) B _(z) S _(z) +g μ _(b)b ₁ S _(z).   (2)

The relaxation rates of the excited triplet and singlet states, shown inFIG. 5, can play a role in the optical pumping and spin measurement. Thetotal magneto-optical response can be modeled using a master equationapproach, such as for example:

$\begin{matrix}{\rho = {{\frac{1}{ih}\left\lbrack {H_{gs}^{1},\rho} \right\rbrack} + {\sum\limits_{k}{L_{k}L_{k}^{\dagger}}} - {\frac{1}{2}L_{k}^{\dagger}L_{k}\rho} - {\frac{1}{2}\rho \; L_{k}^{\dagger}L_{k}}}} & (3)\end{matrix}$

where ρ is the density operator for the NVC ground, excited triplet, andeffective singlet states. The jump operators, L_(k), can have magnitudescorresponding to relaxation rates √{square root over (T_(k))}. Thesolution to the equation can yield the total magneto-optical responsefor both continuous and pulsed excitation.

A first exemplary embodiment of a technique for providing a frequencystandard using spin states of a defect in diamond will now be describedin detail, for purposes of illustration and not limitation, withreference to FIG. 2. In connection with this description, reference willbe made to FIG. 1 for purposes of illustration and not limitation, asone of ordinary skill in the art will appreciate that certain variationsto the system depicted therein can be used.

This exemplary technique, which can be referred to as a “ContinuousWave” (CW) technique, can include continuously applying an initial,adjustable microwave field to a diamond structure concurrently withoptical excitation to produce a detectable photoluminescent response. Inthis embodiment, a diamond chip containing NVCs 110 on a substrate 119can be exposed to optical emissions and exposed to microwave emissionsconcurrently and continuously. That is, an optical beam can be generated(201) and directed (205) to the diamond structure. Concurrently, amicrowave field can be generated (203) and directed (206) to the diamondstructure. The optical beam and microwave field can be generated and canhave characteristics as described herein above in connection with FIG.1.

Under continuous excitation by a microwave field of intensityΩ=gμ_(b)b₁, detuned from resonance of the NVC by an amount Δ=D_(gs)−ω, abroad, phonon-assisted PL response can occur following the equation:

$\begin{matrix}{{{PL}:{F\left( {I,\Omega,\Delta} \right)}} = {{\gamma\rho}_{22}^{ss} + {\frac{\gamma^{2}}{k + \gamma}{\rho_{33}^{ss}.}}}} & (4)\end{matrix}$

Here ρ₂₂ ^(ss) and ρ₃₃ ^(ss) can represent the population of the firstexcited state spin sublevels in the steady-state, as analyticallyderived from Equation 3. FIG. 5( b) shows an exemplary response for Ffor varied detunings, which can be approximated by the Lorentzian:

$\begin{matrix}{{F(\Delta)} = {{I_{0}\left( {1 - \frac{C\; \delta \; v^{2}}{\left( {\Delta/\pi} \right)^{2} + {\delta \; v^{2}}}} \right)}.}} & (5)\end{matrix}$

The PL response can be detected (211) using a photodetector 140. Thephotodetector can be, for example, a silicon photodiode placed adjacentto the diamond structure and opposite the light source. The photodiodecan measure the intensity of the PL response and reports the intensityto the microprocessor 150.

By calculating the PL response, the phase shift A between the microwavefield and D_(gs) can be determined (215) by the processor 150 by usingEquation 5. The frequency of the microwave field can be varied (221) bythe controller 160 until the phase shift is determined to be below thepredetermined threshold (213). The frequency of the microwave field atthat point 153 can be used as a frequency standard (220).

The stability of the frequency standard can be given by a derivationfrom the resonance curve by considering the Allan variance:

$\begin{matrix}{{\sigma_{y}(\tau)} = {\frac{1}{2\pi \; Q}\frac{1}{\left( {S/N} \right)}\frac{1}{\sqrt{\tau}}}} & (6)\end{matrix}$

where S/N is the signal to noise ratio, which can depend on both thephoton shot noise as well as the imperfect modulation of the resonance(e.g., C≠1), and x is the averaging time. The intrinsic linewidth can belimited by the paramagnetic and nuclear spin environments whichfluctuate during the measurement. This linewidth can broaden if themicrowave and optical transitions are driven near saturation; however,higher pump powers can also increase the depth of the dip (C→1). Farbelow optical saturation, the PL rate can be sufficiently small, and themodulation depth (C) reduced, so as to cause a decrease in the stabilityper averaging time. A condition can exist which balances line broadeningwith the reduced signal, and in accordance therewith, the linewidth canbe approximately 3.6 MHz (T₂*=88 ns), and off-resonance fluorescencerate can be approximately 9400 photon/s (accounting for a finitedetector efficiency), and a 17% modulation depth. With these parameters,the Allan variance can be, for example, σ_(y)(τ)=8.124×10⁻⁵τ^(−1/2) fora single NVC. That is, the laser excitation can be reduced far belowsaturation so that optical power broadening can reach the homogeneouslinewidth. At such low pump powers, the fluorescent photon flux can besmall such that the gains in Q-factor can be reduced by losses in signalto noise ratio S/N.

A second exemplary embodiment of a technique for providing a frequencystandard using spin states of a defect in diamond will now be describedin detail, for purposes of illustration and not limitation, withreference to FIG. 3 and FIG. 6. In connection with this description,reference will be made to FIG. 1 for purposes of illustration and notlimitation, as one of ordinary skill in the art will appreciate thatcertain variations to the system depicted therein can be used.

This exemplary technique, which can be referred to as a “pulsed”technique, can include continuously applying a pulsed microwaveexcitation scheme, which can monitor transient fluorescence behavior. Inthis embodiment, optical pumping of the NVC can first prepare theinitial state |ψ₀

=|m₅=0

. This optical pumping can cease before microwaves are emitted. That is,an optical pulse can be generated (301) and directed to the diamondstructure.

A modified Hahn echo sequence can be used to yield a PL response andeliminate the effects of external magnetic fields on the spin response.First, a microwave pulse with a frequency close to 2.870 GHZ and a flipangle of π/4 around the x-axis of the NVC can be generated (310). Afterwaiting (311) a period T (which can be, for example, about I ms), asecond microwave pulse can be generated (313).

The time period T can be, for example, close to the electron spincoherence time, can be approximately 1 ms for high-purity diamondsamples. This second pulse can have a flip angle of π. After waiting(312) a second period of time T, a third microwave pulse can begenerated (315). This third pulse can be identical to the first pulsewith a flip angle π/4. This modified Hahn echo sequence can extend thecoherence time T_(c) (which in the diamond NV can be approximately tensof microseconds) to approximately T₂ by, for example, removing slowlyvarying magnetic fields. In this manner, the modulation of the PLresponse can persist, as the Spin-1 nature of the NVC can allow for a PLsignal proportional to the frequency drift. It should be noted that aconventional Hahn echo sequence can remove the phase accumulationassociated with frequency drift with the additional π pulse therein.

Evolution under this drift echo sequence, U_(echo), can follow theequation

$\begin{matrix}{\left. {{\psi_{f}\rangle} = {{U_{echo}{\psi_{0}\rangle}} = {{\frac{1}{\sqrt{2}}{\sin (\varphi)}{0\rangle}} - {\frac{1}{2}{\cos (\varphi)}\left( {{+ 1}} \right)} + {{- 1}\rangle}}}} \right),} & (7)\end{matrix}$

where φ=(D_(gs)−w)T=δwT. A transient fluorescence measurement can berecorded (305) immediately following the third pulse. For example, theresponse can be measured by recording the photocurrent with thephotodetector 140 for about 300 ns timed with a pulse of green light(303) from the light source 120. In certain embodiments, the timing canbe shorter than the electron spin reset time, which can be for examplebetween approximately 200 ns and approximately 500 ns.

The transient PL response of the NVC can be modeled, for example, usingprojective measurements. The operator M can describe the spinexpectation value for a PL measurement, M=α|0

0|+b(|+1)<+1|+|−1)<−1|), where a and b are independent Poisson randomvariables. The fractional frequency deviation can vary as the quantumobservable, M for the state |ω_(f)> according to:

$\begin{matrix}{{\frac{\delta \; w}{w_{0}} = {\frac{1}{w_{0}}\frac{\langle{\Delta \; \hat{M}}\rangle}{w_{0}{{{\partial{\langle\hat{M}\rangle}}/{\partial w}}}}}},} & (8)\end{matrix}$

where

Δ{circumflex over (M)}²>=<{circumflex over (M)}²>−<{circumflex over(M)}>² is the variance of th operator. For room temperature spin readoutof the NVC, for example, 2a=3b. By calculating the moments of{circumflex over (M)} and assuming the results for total time τ=M′T areaccumulated, the following equation can be derived:

$\begin{matrix}{{{\langle\frac{\delta \; w}{w_{0}}\rangle}_{M^{1}} = \frac{\xi}{D_{gs}\sqrt{T\; \tau}}},} & (9)\end{matrix}$

with ξ≈5 due to a combination of imperfect spin readout (i.e. b≠0),inefficient collection of photons (i.e. a<<1), and a small ratio ofshelving state to radiative lifetime (λ/γ).

The predetermined threshold for the phase shift for an ensemble of NVCscan be predetermined, for example, by multiplying the deviation for asingle NVC, 8.8×10⁻⁹/√τ by 1/√N, where N is the number of NVCs. N can beestimated by taking the density of pure diamond, 1.74×10²³ C atoms/cm³,with the NV′ defect fraction at 0.01 parts per billion (ppb), whichresults in ˜1.74×10¹² NVCs/cm³. For a diamond chip of cross-section areaof 1 mm³ and a thickness of approximately 100 μm, this can result in adeviation of 2×10⁻¹³/√τ in accordance with this exemplary technique.Thus, a diamond film with thickness of approximately 100 μm can give anAllen variance σ_(y) ^(pulsed)˜6.7×10⁻¹³ τ^(1/2).

If the determined phase shift is above the predetermined threshold(320), the microwave frequency can be adjusted (317) by the controller160 and the pulse sequence can be repeated. If the detellnined phaseshift is below the predetermined threshold (320), then the frequency ofthe microwave field determined by the processor 153 can be used as thefrequency standard (330).

A third exemplary embodiment of a technique for providing a frequencystandard using spin states of a defect in diamond will now be describedin detail, for purposes of illustration and not limitation, withreference to FIG. 7. In this embodiment, techniques for obtaining afrequency standard from the crystal field splitting frequency of anitrogen vacancy center in a diamond structure can include usingtemperature stabilization techniques to improve the accuracy of thefrequency standard.

The resonance frequency corresponding to D_(gs) of an NV center can varyas a function of temperature, θ. At room-temperature, dD_(gs)/dθ can beequal to approximately 74.2 kHz/K for mm-sized samples. At temperaturesof approximately 5K, dD_(gs)/dθ can be equal to approximately 100 kHz/K,and at higher temperatures (up to approximately 600K), dD_(gs)/dθ can beequal to approximately 100 kHz/K. Thus, zero-field splitting temperaturedependence can be nonlinear. This temperature dependence can be a resultof thermal expansion of the diamond. For example, local lattice spacingof the NV center can be distorted, causing changes in the orbitaloverlaps which determine D_(gs).

The temperature of the diamond chip can be stabilized, for example,using commercially available Peltier coolers with PID loops (for exampleto within 0.01K). This can allow, for example, 742 Hz uncertainty in thezero-field splitting, or a fraction frequency stability of approximately2.58×10⁻⁷ at room temperature. However, in this case, ensemble averagingwould not increase certainty, as all NV centers can shift equally due toisotropic expansion.

In accordance with this exemplary embodiment, anisotropy can be inducedin the crystal's temperature response. Two different dependences can beidentified, and can be locked in a feedback loop to enhance temperaturestability. In this embodiment, two or more “clocks” (that is, thesubject matter described herein), can have different temperaturedependences. Different temperature dependences can be achieved, forexample, by utilizing two distinct diamond slabs mounted on substrateswith different thermal expansion coefficients.

For example, and with reference to FIG. 7, two diamond chips 815 and 825can be clamped into two different substrates 810 and 820. The substratescan have different thermal expansion coefficients. One diamond chip 815can be clamped in a stiff material 810 with Young's modulus E_(t) and ahigh thermal expansion coefficient η_(c1). This material can be, forexample, brass. The second diamond chip 825 can be clamped in adifferent material 820 with Young's modulus E₂ and a lower theiivalexpansion coefficient η_(c2) where η_(c2)<η_(c1). The second materialcan be, for example, tungsten. The cross-sectional area of thesubstrates can be much greater than that of the diamond chips. Bothdiamond chips and the substrates can be assumed to start at the sametemperature θ. The change in strain imparted on the diamond can beapproximated as Δε_(1,2)≈η_(d)(1+η_(c1,2)E_(c1,2)/E_(d)) Δθ, where Δθ isthe temperature difference between the initial set-point T₀. Thisinitial set-point can be determined at a point where the two clocks havethe same initial frequency ω₀ and can be adjusted by pre-loading strainwithin the samples, as seen in FIG. 7( a).

A conventional thermal feedback system using thermistors can maintainthe system near T₀ within ˜10⁻²-10⁻³ K. To compensate for smallertemperature drifts, the different temperature dependencies of the twodiamond chips can be exploited to yield the equations:

ω₁(T)=ω₀+β₁Δθ  (10)

ω₂(T)=ω₀+β₂Δθ  (11)

where

$\beta_{1,2} = {{\frac{ɛ}{\theta}\frac{D_{gs}}{ɛ}} = {{\eta_{d}\left( {1 + {\eta_{{c\; 1},2}{E_{{c\; 1},2}/E_{d}}}} \right)}{\left( {{D_{gs}}/{ɛ}} \right).}}}$

At time τ=0, both clocks can be at the same temperature θ₀, before thetemperature is allowed to fluctuate within a small range around θ₀.After time t, these clocks have acquired a phase:

φ_(1,2)(t)=ω₀ t+∫ ₀ ¹β_(1,2), Δθ(t)dt′±Δφ ₀   (12)

where Δφ₀=ξ√{square root over (t)}/°{square root over (θ_(2,ens)N)} isthe phase uncertainty. The difference between the two phases can berecorded by mixing and low-pass filtering the two clock signals, givingΔφ(t)=φ₂(t)−φ₁(t)=∫₀ ¹Δβ_(1,2)Δθ(t′)dt′±√{square root over (2)}Δφ₀,where β_(1,2)=β₂−β₁ and an identical variance is assumed for φ₁(t) andφ₂(t).

The temperature fluctuations in clock ‘1’ can be corrected as follows:

$\begin{matrix}\begin{matrix}{{\varphi_{1}^{\prime}(t)} = {{\varphi_{1}(t)} - {\int_{0}^{t}{\beta_{1}\Delta \; {T\left( t^{\prime} \right)}{t^{\prime}}}}}} \\{= {{\varphi_{1}(t)} - {\left( {\int_{0}^{t}{\beta_{1,2}\Delta \; {T\left( t^{\prime} \right)}{t^{\prime}}}} \right) \cdot \frac{\beta_{1}}{{\Delta\beta}_{1,2}}}}} \\{= {{\varphi_{1}(t)} - {\frac{\beta_{1}}{{\Delta\beta}_{1,2}}\left( {{{\Delta\varphi}(t)} \mp {\sqrt{2}{\Delta\varphi}_{0}}} \right.}}}\end{matrix} & (13)\end{matrix}$

The phase can be divided by t to yield the frequency, from which theuncertainty of the new “synchronized composite” after a time t:

$\begin{matrix}\begin{matrix}{\frac{{\Delta\omega}_{1}^{\prime}}{\omega_{1}^{\prime}} = {\frac{\xi}{\sqrt{T_{2}{Nt}}D_{gs}}\left( {1 + {2\left( \frac{\beta_{1}}{{\Delta\beta}_{1,2}} \right)^{2}}} \right)^{1/2}}} \\{= {\left( \frac{\Delta\omega}{\omega_{0}} \right)_{T = T_{0}}{\left( {1 + {2\left( \frac{\beta_{1}}{{\Delta\beta}_{1,2}} \right)^{2}}} \right)^{1/2}.}}}\end{matrix} & (14)\end{matrix}$

Thus, the uncertainty can be reduced when the temperature dependenciesof the two clocks can be large. For example, with reference to equation14, when Δβ₂ >>β₁, the performance of the composite clock can be similarto that of a bare temperature-insensitive NV clock. While the phasedifference can be directly computed and the clock frequency correcteddigitally, the phase difference can also be used to stabilize thetemperature, particularly where the two clocks can be maintained at afrequency difference, v_(beat)˜10 kHz, so that the beat frequency can belocked to a high Q (˜10⁶) quartz oscillator.

In another exemplary embodiment, the temperature dependence can beresolved in a single diamond system through the use of engineeredstrain. Within the ground state Hamiltonian, as expressed in equation 1,the strain dependence can be hidden in the effective electric fieldvector, {right arrow over (σ)}. In terms of the actual components of thestrain tensor ε, perturbation to the spin Hamiltonian can take the formF_(ijkl)S_(i)S_(j)ε_(kl), where F is the forth order strain responsetensor. Symmetries can reduce the number of allowed terms to eighttotally symmetric combinations, and of these, only the terms with anS_(z) ² coefficient can be considered. Thus, the strain dependence ofthe zero-field splitting can generally be characterized as:

(D _(gs) +A ₁(ε_(xx)+ε_(yy))+A ₂ε_(zz))(S _(z) ²−2/3),   (14)

where A₁ and A₂ are parameters which can require experimental input. Inthe case of isotropic expansion, ε_(xx)=ε_(yy)=ε_(zz) dD_(gs)/dθ can beequal to 2A₁+A₂. By clamping the diamond along a specific director, ananisotropic lattice response can significantly reduce the effectivetemperature dependence of the zero-field splitting.

For example, with reference to FIG. 7 and FIG. 8, a small temperatureshift Δθ, near θ₀, can cause a diamond slab of length L_(d) to expand byΔL=L_(d)η_(d)Δθ. If the diamond slab were instead clamped, thisexpansion can be modulated. For example, Material 2 830, which forms thebottom of the clamp, can have a low thermal expansion coefficient and ahigh Young's modulus. Material 1 832, which forms the sides of theclamp, can have a high thermal expansion coefficient and a high Young'smodulus. With varying temperature, a pressureP=E_(d)ΔL_(d)/L=E_(d)ε_(d)=E_(d)ΔTη_(d) can be exerted to compress thediamond chip 835 by −ΔL. For temperature changes 0.01K, this pressure isapproximately 12.2 kPa.

In connection with this embodiment, the different temperaturedependences of NVCs in different orientations within the diamond chipcan be used. NVCs can have one of four directions within the crystalstructure. Anisotropic strain can be induced in the diamond chip, forexample by patterning slits into the diamond chip. For example, theslits may be patterned using optical or electron beam lithography,followed by plasma etching with gasses such as oxygen or chlorine.

The single diamond chip system can be measured by creating a microwavefield to measure the transient photoluminescence response of NVCs in oneorientation, then measuring the response of NVCs in another orientationthat will have a different temperature dependence.

The foregoing merely illustrates the principles of the disclosed subjectmatter. Various modifications and alterations to the describedembodiments will be apparent to those skilled in the art in view of theteaching herein. It will thus be appreciated that those skilled in theart will be able to devise numerous techniques which, although notexplicitly described herein, embody the principles of the disclosedsubject matter and are thus within the spirit and scope of the disclosedsubject matter.

1. A method for obtaining a frequency standard using the crystal fieldsplitting frequency of at least one nitrogen vacancy center in at leastone diamond structure, comprising: applying a microwave field to the atleast one diamond structure, the microwave field having an adjustableand an initial frequency; optically exciting the at least one diamondstructure to create photoluminescence having a photoluminescentcharacteristic based at least on the frequency of the microwave field;detecting at least a first measurement of the photoluminescentcharacteristic at the initial microwave frequency; determining, based onthe first measurement, a phase shift for the known microwave frequencyfrom the constant crystal field splitting frequency of the at least onediamond structure; varying the frequency of the microwave field untilthe phase shift is below a predetermined threshold; and using thefrequency of the microwave field as a frequency standard.
 2. The methodof claim 1, wherein the frequency of the microwave field is continuouslymonitored to prevent drift of the frequency standard.
 3. The method ofclaim 1, further comprising surrounding the at least one diamondstructure with a dielectric cavity prior to applying the microwave fieldto allow optical excitation at a lower output power.
 4. The method ofclaim 1, wherein the optically exciting further comprises opticallypumping the at least one diamond structure.
 5. The method of claim 4,wherein detecting at least a first measurement further comprisescontinuously detecting measurements.
 6. The method of claim 1, whereinthe optically exciting further comprises continuously optically pumpingthe at least one diamond structure.
 7. The method of claim 1, whereinthe optically exciting further comprises applying pulsed emissions toexcite the at least one nitrogen vacancy center.
 8. The method of claim7, wherein the at least one nitrogen vacancy center is optically excitedprior to the microwave pulse emissions.
 9. The method of claim 7,further comprising: optically exciting the at least one nitrogen vacancycenter with a pulse of light having a wavelength of 532 un; applying an/4 microwave pulse; waiting a time of period T; applying a pi microwavepulse; waiting a second time of period T; applying a π/4 microwave pulsein the same phase as the first π/4 microwave pulse; optically excitingthe at least one nitrogen vacancy center with a pulse of light having awavelength of 532 nm; measuring the transient fluorescence response ofthe at least one nitrogen vacancy center; and determining, from thetransient fluorescence response, the phase shift of the appliedmicrowave pulse from the crystal field splitting frequency of the atleast one nitrogen vacancy center in the at least one diamond structure.10. The method of claim 1, wherein the at least one diamond structurecomprises a first and a second diamond structures, each having at leastone nitrogen vacancy center, and wherein the first measurement detectingfurther comprises simultaneously detecting a first measurement for thefirst diamond structure and a first measurement for the second diamondstructure, to thereby stabilize the detected measurements with respectto temperature.
 11. The method of claim 1, the at least one nitrogenvacancy having a temperature dependence, and further comprisingcontrolling the temperate dependence of the at least one nitrogenvacancy center by engineered strain.
 12. A system for obtaining afrequency standard using the crystal field splitting frequency of atleast one nitrogen vacancy center in at least one diamond structure,comprising: a dielectric cavity adapted to at least partially encompassthe at least one diamond structure; a light source, operativelyconfigured to excite the at least one nitrogen vacancy, thereby allowingthe at least one nitrogen vacancy center to produce a photoluminescentresponse; a photodetector disposed proximal to the dielectric cavity andopposite the light source for detecting photoluminescence; a stripline,disposed in the plane of and at least partially encompassing the atleast one diamond structure, for applying microwave emissions to the atleast one diamond structure; a processor, coupled to the photodetectorand the stripline, to calculate the phase shift of the frequency of themicrowave emissions from the crystal field splitting frequency of the atleast one nitrogen vacancy center in the at least one diamond structure;and a controller, coupled to the processor and the stripline, to adjustthe frequency of the microwave emissions from the stripline based on thedetermined frequency from the processor.
 13. The system of claim 12,wherein the photodetector comprises a silicon photodiode.
 14. The systemof claim 12, wherein the dielectric cavity comprises a pair of Braggreflectors to create a 532 nm resonant cavity.
 15. The system of claim14, wherein the pair of Bragg reflectors comprises one reflector placedon one face of the at least one diamond structure outside the plane ofthe stripline for applying microwave emissions, and further comprisesthe second Bragg reflector placed on the opposite face of the at leastone diamond structure.
 16. The system of claim 12, wherein the lightsource comprises a surface emitting laser adapted for generation of adoubled 1064 nrn beam.
 17. The system of claim 12, wherein the lightsource comprises a laser adapted to continuously irradiate the at leastone diamond structure.
 18. The system of claim 17, wherein the striplinecomprises a microwave source adapted to apply microwave emissionscontinuously upon the at least one diamond structure.
 19. The system ofclaim 12, wherein the light source comprises a laser adapted forgeneration of pulses for application to the at least one diamondstructure.
 20. The system of claim 19, wherein the stripline comprises amicrowave source adapted for generation of pulsed emissions upon the atleast one diamond structure.
 21. The system of claim 12, wherein the atleast one diamond structure comprises a first and a second diamondstructures, each having at least one nitrogen vacancy center, whereinthe processor is further adapted for simultaneously detecting a firstmeasurement for the first diamond structure and a first measurement forthe second diamond structure, to thereby stabilize the detectedmeasurements with respect to temperature.
 22. The system of claim 21,further comprising a first substrate having a first thermal expansioncoefficient and a second substrate having a second thermal expansioncoefficient, wherein the first diamond structure is coupled to the firstsubstrate and the second diamond structure is coupled to the secondsubstrate, and wherein the first and second thermal expansioncoefficients are different.
 23. The system of claim 12, wherein the atleast one diamond structure is coupled to a single clamp to therebyinduce strain thereon.